5 Alex and Beth play a zero-sum game. Alex chooses a strategy P, Q or R and Beth chooses a strategy \(\mathrm { X } , \mathrm { Y }\) or Z . The table shows the number of points won by Alex for each combination of strategies. The entry for cell \(( \mathrm { P } , \mathrm { X } )\) is \(x\), where \(x\) is an integer.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Beth}
| | X | Y | Z |
| \cline { 3 - 5 } | P | \(x\) | 3 | 2 |
| \cline { 3 - 5 } | Q | 4 | 0 | - 2 |
| \cline { 3 - 5 } | R | - 3 | - 1 | - 3 |
| \cline { 3 - 5 } | | | | |
| \cline { 3 - 5 } |
\end{table}
Suppose that P is a play-safe strategy.
- Determine the values of \(x\) for which the game is stable.
- Determine the values of \(x\) for which the game is unstable.
The game can be reduced to a \(2 \times 3\) game using dominance.
- Write down the pay-off matrix for the reduced game.
When the game is unstable, Alex plays strategy P with probability \(p\).
- Determine, as a function of \(x\), the value of \(p\) for the optimal mixed strategy for Alex.
Suppose, instead, that P is not a play-safe strategy and the value of \(x\) is - 5 .
- Show how to set up a linear programming formulation that could be used to find the optimal mixed strategy for Alex.